Problem: How many perfect cubes are between 100 and 900?
Answer: The smallest perfect cube inside this range is $5^3 = 125$ since $4^3 = 64.$  As for the highest cube, we know that $10^3 = 1000,$ which is outside the range, so we try $9^3 = 729.$  Therefore the cubes in question are $5^3,6^3,7^3,8^3,9^3$.  So there are $\boxed{5}$ such cubes.